The Obstruction to Fibering a Manifold over a Circle

Abstract

Let M be a compact, connected, smooth manifold whose dimension is greater than five, and let / be a continuous map from M to the circle, which we denote by S. Suppose that / restricted to the boundary of M9 denoted by bM9 is a smooth fibration. We note that a map h from a smooth manifold N to S is a smooth fibration if h is smooth, and for each point x of N the derivative of h maps the tangent plane to N at x onto the tangent plane to S at h(x). We wish to address ourselves in this talk to the following problem.